How Does the Wave Function Behave Across a Step Potential in Quantum Mechanics?

[abubabu]

A particle with mass m and energy E is moving in one dimension from right to left. It is incident on the step potential V(x)=0 for x<0 and V(x)=Vo <--(pronounced "v not") for x>=0, where Vo>0, as shown on the diagram. The energy of this particle is E>Vo.
(The diagram has a particle coming in from the left above V=Vo)

Solve the Schrödinger equation to derive the wave function for x<0 and x>=0. Express the solution in terms of a single unknown constant.

I have been looking at the case when a particle is coming in from the right and E>Vo, but I am failing to make a connection between the two. I really need the idea explained to me here, I'm missing something!

 

[monish]

I just wrote up an answer to almost the same question in another thread with almost the same name as this one: "1-dimensional time-independent Schroedinger equation"

The question in the other thread was messed up because it put in a second step potential; but essentially you didn't need to worry about that in the end. Here is my write-up from the other thread:

I can't read the correction you made to your original post but the problem only makes sense to me if there is an incoming particle (wave actually) from the left, a step in potential at x=0 and another one at x=a. Then there is a reflection at the first boundary, a middle zone with waves going both ways, and a transmitted beam at the second boundary.

I think you already have expressions for the waves, which is to say their k numbers, based on the three potentials. So there are 5 undetermined coefficients, in general complex, for the wave amplitudes.

However, as I see it, it gets a lot simpler when you consider what they are asking for: just the ratio between the left- and right- propagating waves in the middle zone. Let's take the outgoing wave in zone 3 to have unit amplitude; in fact, make the wave number something simple like 1 so it is just:

exp(ikx - wt)

(of course we won't worry to much about the wt).

Then you just need to match up the waves in zone 2, let's give them a wave number like 3 or something:

Aexp(i3x) + Bexp(-i3x)

And I believe the boundary condition is that both the amplitudes and their derivatives have to match up at the transition. With the arbitrary numbers I put in above, I can solve pretty easily: I get A + B = 1 for the amplitudes, and 3A - 3B = 1 for the derivatives.

So it seems I can solve the problem as stated without even worrying about what happened at the first boundary, with the incoming particle. Does this look right?

Marty